Regularized Upper Incomplete Gamma Functions

In summary, the conversation is discussing the possibility of finding a property or function, denoted as X, that relates to adding two upper incomplete regularized gamma functions with positive half-integer values. The extra information provided suggests that it may not be possible to find such a function, as it would depend on specific values of z1, z2, and a.
  • #1
natski
267
2
Does anyone know of any properties in relation to adding two upper incomplete regularized gamma functions?

i.e. can you write

Q(a,z1) - Q(a,z2) = Q(a,X) ?
where X(z1,z2)

Extra information: a is a positive half-integer.

Natski
 
Last edited:
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  • #2
If anyone feels that this problem is not solvable then please say so too!
 
  • #3
natski said:
Does anyone know of any properties in relation to adding two upper incomplete regularized gamma functions?

i.e. can you write

Q(a,z1) - Q(a,z2) = Q(a,X) ?
where X(z1,z2)

Extra information: a is a positive half-integer.

Natski


I feel it will not be possible to find such an X, if it should be a function of z1 and z2 only. Of course for each possible z1,z2,a you can define X as the number to satisfy your equation, but this is probabably not what you want.:smile:
 

1. What is a regularized upper incomplete gamma function?

A regularized upper incomplete gamma function, denoted as Γ*(a,x), is a mathematical function that is used to calculate the probability of a random variable having a value greater than or equal to x, given that the variable follows a gamma distribution with shape parameter a.

2. How is the regularized upper incomplete gamma function different from the upper incomplete gamma function?

The regularized upper incomplete gamma function is a normalized version of the upper incomplete gamma function, which means it takes into account the total area under the gamma distribution curve. The regularized function is always between 0 and 1, while the upper incomplete gamma function can have values greater than 1.

3. What is the relationship between the regularized upper incomplete gamma function and the cumulative distribution function (CDF) of the gamma distribution?

The regularized upper incomplete gamma function is essentially the complement of the CDF of the gamma distribution. This means that Γ*(a,x) = 1 - CDF(a,x), where CDF(a,x) is the probability of a random variable with shape parameter a being less than or equal to x.

4. How is the regularized upper incomplete gamma function used in statistics?

The regularized upper incomplete gamma function is often used in statistics to calculate confidence intervals for the mean of a gamma-distributed population. It is also used in the analysis of survival data and reliability analysis, among other applications.

5. Are there any special properties or formulas associated with the regularized upper incomplete gamma function?

Yes, there are several special properties and formulas associated with the regularized upper incomplete gamma function, including the recurrence relation, series representation, and integral representation. It also has connections to other mathematical functions, such as the incomplete beta function and the error function.

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